The Commutators of Classical Groups

In this paper, we compare the decoding error rates in the error floors for non-binary low-density parity-check (LDPC) codes over the general linear group with those for non-binary LDPC codes over finite field transmitted over the q-ary memoryless symmetric channel under belief propagation decoding. To analyze non-binary LDPC codes defined over both general linear group GL(m, F 2 ) and finite field F 2m , we investigate non-binary LDPC codes defined over GL(m 3 , F 2m4 ). We propose a method to lower the error floors for non-binary LDPC codes. In this analysis, we see that the optimized non-binary LDPC codes defined over general linear group have the same decoding performance in the error floors as those defined over finite field. The non-binary LDPC codes defined over general linear group have more choices of the labels in the edges which satisfy the condition for the optimization.

In this article we give a branching rule for Harish–Chandra restriction from the general linear group Gl n (q) to the Levi subgroup Gl n−1(q) × Gl1(q) in the case of the unipotent block.

The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received much substance and impetus from the work of Severi on commutative algebraic groups over the complex number field, that of Kolchin, Chevalley, and Borel on algebraic groups of matrices, and especially Weil's research on abelian varieties and algebraic transformation spaces. The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic groups of matrices, or actually of neither of these types.

Let Γ be a connected reductive group over an algebraically closed field F of positive characteristic and let G be the set of fixed points of Γ under some Frobenius map of Γ. Then G is a finite group of Lie type. The p-modular representations of G for p = char F (the “describing” characteristic case) are closely related to rational representations of Γ, and thus results from the theory of reductive algebraic groups can be used to develop the representation theory of finite groups of Lie type in the describing characteristic case. One of the outstanding open questions in this area is to determine the dimensions of the irreducible representations and the decomposition matrix whose entries record multiplicities of irreducible modules as composition factors of Weyl modules. One way to define Weyl modules is by reducing modulo p (p = char F) the irreducible modules of the complex algebraic group of the same type. Here reduction modulo p happens in both, in the field underlying the group and in the field over which the representing matrices are defined, simultaneously. One of the main tools to study these questions is derived from the Weyl group of Γ. If, for example, Γ is a general linear group, the decomposition matrix of the Weyl modules can be calculated from decomposition numbers of Schur algebras, which in turn can be calculated in terms of symmetric groups. We shall discuss this phenomenon in more detail below.

Structure Arising from Induction and Jacquet Modules of Representations of Classical p-Adic Groups

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In previous papers, nice trinomial equations were given for unramified coverings of the once punctured affine line in nonzero characteristic p p with the projective general group P G L ( m , q ) \mathrm {PGL}(m,q) and the general linear group G L ( m , q ) \mathrm {GL}(m,q) as Galois groups where m > 1 m>1 is any integer and q > 1 q>1 is any power of p p . These Galois groups were calculated over an algebraically closed ground field. Here we show that, when calculated over the prime field, as Galois groups we get the projective general semilinear group P Γ L ( m , q ) \mathrm {P}\Gamma \mathrm {L}(m,q) and the general semilinear group Γ L ( m , q ) \Gamma \mathrm {L}(m,q) . We also obtain the semilinear versions of the local coverings considered in previous papers.

We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from mathbf {PSL}_n(q) collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is mathbf {PSp}_{2n}(q), mathbf {P}{varvec{Omega }}^+_{4n}(q), mathbf {P}{varvec{Omega }}^-_{4n}(q), ^3D_4(q), E_7(q), E_8(q), F_4(q), or G_2(q) with q even is the group algebra.

This is the first in a series of papers dedicated to the structure of Chevalley groups over commutative rings. The goal of this series is to systematically develop methods of calculations in Chevalley groups over rings, based on the use of their minimal modules. As an application, we give new direct proofs for normality of the elementary subgroup, description of normal subgroups and similar results due to E. Abe, G. Taddei, L. N. Vaserstein, and others, as well as some generalizations. In this first part we outline the whole project, reproduce construction of Chevalley groups and their elementary subgroups, recall familiar facts about the elementary calculations in these groups, and fix a specific choice of the structure constants.

The realization of a BRST cohomology of the 4D topological gauge-affine gravity is established in terms of a superconnection formalism. The identification of fields in the quantized theory occurs directly as is usual in terms of superconnection and its supercurvature components with the double covering of the general affine group GA¯(4,R). Then, by means of an appropriate decomposition of the metalinear double-covering group SL¯(5,R) with respect to the general linear double-covering group GL¯(4,R), one can easily obtain the enlargements of the fields while remaining consistent with the BRST algebra. This leads to the descent equations, allowing us to build the observables of the theory by means of the BRST algebra constructed using a sa¯(5,R) algebra-valued superconnection. In particular, we discuss the construction of topological invariants with torsion.

The article is devoted to the classification of the minimal and minimal nontrivial inductive systems of modular representations for naturally embedded algebraic and finite groups of type A and related locally finite groups. It occurs that the minimal systems consist of the trivial representations for the relevant groups and the minimal nontrivial ones are connected with Frobenius twists of the standard representations and their duals. These results are applied to the description of the maximal ideals in group algebras of the locally finite groups SL ∞ and SU ∞ in describing characteristic.

Let A be an algebra with involution ∗ over a field F. We recall that A is said to be normal if xx∗ = x∗x for every x ∈ A. Over the decades, normal algebras with involutions have been extensively investigated on their own (see e.g. [1], [3], [4], [5], [7], [8], [9], [10], [11]) and, moreover, they have several applications in linear algebra and functional analysis as well (see e.g. [2], [6], [11], [12], [13]). It is well-known that any normal algebra with involution satisfies the standard polynomial identity of degree 4 (see [7, §5]). Moreover, in [11] Maxwell determined the structure of a normal simple algebra of matrices with entries in a field with involution. He also proved that a division algebra D with involution is normal if and only if D is either a field or a generalized quaternion algebra over its center. Furthermore, a characterization of group algebras which are normal under the standard involution was established by Bovdi, Gudivok, and Semirot in [3]. Subsequently, such a result has been extended to twisted group algebras in [4] and to group algebras under a Novikov involution in [5]. On the other hand, it seems that the rather natural problems of characterizing ordinary and restricted enveloping algebras which are normal under their canonical involutions have not been settled yet. The present paper is just devoted to answering these questions. For an arbitrary Lie algebra L we denote by U(L) the universal enveloping algebra of L. Moreover, if L is restricted with a p-map [p] over a field F of characteristic p > 0, then we denote by u(L) the restricted enveloping algebra of L. We consider U(L) and u(L) with the principal involution ∗, namely, the unique F-antiautomorphism such that x∗ = −x for every x in L. Note that ∗ is just the antipode of the F-Hopf algebras U(L) or u(L). We use the symbols Z(L) and L′ for the center and the derived subalgebra of L, respectively. If S ⊆ L then we denote by 〈S〉F and 〈S〉 the F-vector space and the subalgebra generated by S. Also, if L is restricted then 〈S〉p denotes the restricted subalgebra generated by S and we put S = {x |x ∈ S}. In our first main result we completely settle the restricted case:

In this paper we deduce a graded version of Quillen–Suslin’s Local-Global Principle for the traditional classical groups, viz. general linear, symplectic and orthogonal groups and establish its equivalence of the normality property of the respective elementary subgroups. This generalizes previous result of Basu–Rao–Khanna. Then, as an application, we establish an analogue Local-Global Principle for the commutator subgroups of the special linear and symplectic groups. Finally, by using Swan–Weibel’s homotopy trick, we establish graded analogue of the Local-Global Principle for the transvection subgroups of the full automorphism groups of the linear, symplectic and orthogonal modules. This generalizes the previous result of Bak–Basu–Rao.

Let $\\Sigma_{g,1}$ be an oriented compact surface of genus $g$ with 1 boundary component, and $\\Gamma_{g,1}$ the mapping class group of $\\Sigma_{g,1}$. We determine the stable cohomology group of $\\Gamma_{g,1}$ with coefficients in $H^1 (\\Sigma_{g ,1} ; \\mathbb{Z})^{\\otimes n}$, $n \\ge 1$, explicitly modulo the stable cohomology group with trivial coefficients. As a corollary the rational stable cohomology algebra of the semi-direct product $\\Gamma_{g,1} \\ltimes H_1 (\\Sigma_{g,1} ; \\mathbb{Z})$ (which we call the extended mapping class group) is proved to be freely generated by the generalized Morita-Mumford classes $\\widetilde{m_{i,j}}$'s $(i \\ge 0,\\, j \\ge 1,\\, i+j \\ge 2)$ [11] over the rational stable cohomology algebra of the group $\\Gamma_{g,1}$.

Overgroups of symplectic group in linear group over commutative rings